Environmental Engineering Reference
In-Depth Information
Table A.1
Variable
Dimensions
k
=
mass transfer coefficient
L
/
t
v
∞
=
free stream fluid velocity
L
/
t
L
2
ν
=
kinematic viscosity of fluid
/
t
L
2
D
AB
=
solute diffusion coefficient
/
t
L
=
plate length
L
in the set. As an example, with density (
ρ
), viscosity (
µ
) and kinematic viscosity (
ν
), only
two of the three variables are independent. Can you see why?
As an example of the method, let us examine the situation of mass transfer associated
with flow across a flat plate. The variables which are important are listed in Table A.1,
together with their dimensions. In this case,
=
n
−
m
=
5
−
2
=
3.
ν
, which do not form a dimensionless set. This leaves
three variables for use to form the three
Choose two variables, say
L
and
ν
together with each
of the other three variables so that each result is dimensionless. The method is to raise
each variable to the appropriate power so that each dimension sums to zero. Choosing
k
as the variable:
s. We then combine L and
L
a
b
k
L
a
L
2
b
t
−
b
Lt
−
1
ν
=
.
We can write an equation for the sum of the exponents for each dimension. Each equation
must sum to zero for the resulting group to be dimensionless.
L:
a
+
2
b
+
1
=
0
a
=
1
kL
ν
1
=
.
t:
−
b
−
1
=
0
b
=−
1
D
AB
ν
Similarly, using
D
AB
as the other variable gives:
2
=
, and using
v
∞
gives:
3
=
v
∞
L
ν
. We can then write:
f
v
∞
L
ν
kL
ν
D
AB
=
,
.
We can recognize two dimensionless groups as:
=
v
∞
L
ν
D
AB
,
Re
;
Sc
=
where
Re
and
Sc
are known as the Reynolds and the Schmidt numbers, respectively. The
group
kL
ν
kL
D
AB
=
is usually not used,
Sh
being a better choice (
Sh
is the Sherwood
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